Metamath Proof Explorer

Theorem dipsubdi

Description: Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007) (New usage is discouraged.)

Ref Expression
Hypotheses ipsubdir.1 
|- X = ( BaseSet ` U )
ipsubdir.3 
|- M = ( -v ` U )
ipsubdir.7 
|- P = ( .iOLD ` U )
Assertion dipsubdi 
|- ( ( U e. CPreHilOLD /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A P ( B M C ) ) = ( ( A P B ) - ( A P C ) ) )

Proof

Step Hyp Ref Expression
1 ipsubdir.1 
 |-  X = ( BaseSet ` U )
2 ipsubdir.3 
 |-  M = ( -v ` U )
3 ipsubdir.7 
 |-  P = ( .iOLD ` U )
4 id 
 |-  ( ( C e. X /\ B e. X /\ A e. X ) -> ( C e. X /\ B e. X /\ A e. X ) )
5 4 3com13 
 |-  ( ( A e. X /\ B e. X /\ C e. X ) -> ( C e. X /\ B e. X /\ A e. X ) )
6 id 
 |-  ( ( B e. X /\ C e. X /\ A e. X ) -> ( B e. X /\ C e. X /\ A e. X ) )
7 6 3com12 
 |-  ( ( C e. X /\ B e. X /\ A e. X ) -> ( B e. X /\ C e. X /\ A e. X ) )
8 1 2 3 dipsubdir 
 |-  ( ( U e. CPreHilOLD /\ ( B e. X /\ C e. X /\ A e. X ) ) -> ( ( B M C ) P A ) = ( ( B P A ) - ( C P A ) ) )
9 7 8 sylan2 
 |-  ( ( U e. CPreHilOLD /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( ( B M C ) P A ) = ( ( B P A ) - ( C P A ) ) )
10 9 fveq2d 
 |-  ( ( U e. CPreHilOLD /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( * ` ( ( B M C ) P A ) ) = ( * ` ( ( B P A ) - ( C P A ) ) ) )
11 phnv 
 |-  ( U e. CPreHilOLD -> U e. NrmCVec )
12 simpl 
 |-  ( ( U e. NrmCVec /\ ( C e. X /\ B e. X /\ A e. X ) ) -> U e. NrmCVec )
13 1 2 nvmcl 
 |-  ( ( U e. NrmCVec /\ B e. X /\ C e. X ) -> ( B M C ) e. X )
14 13 3com23 
 |-  ( ( U e. NrmCVec /\ C e. X /\ B e. X ) -> ( B M C ) e. X )
15 14 3adant3r3 
 |-  ( ( U e. NrmCVec /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( B M C ) e. X )
16 simpr3 
 |-  ( ( U e. NrmCVec /\ ( C e. X /\ B e. X /\ A e. X ) ) -> A e. X )
17 1 3 dipcj 
 |-  ( ( U e. NrmCVec /\ ( B M C ) e. X /\ A e. X ) -> ( * ` ( ( B M C ) P A ) ) = ( A P ( B M C ) ) )
18 12 15 16 17 syl3anc 
 |-  ( ( U e. NrmCVec /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( * ` ( ( B M C ) P A ) ) = ( A P ( B M C ) ) )
19 11 18 sylan 
 |-  ( ( U e. CPreHilOLD /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( * ` ( ( B M C ) P A ) ) = ( A P ( B M C ) ) )
20 1 3 dipcl 
 |-  ( ( U e. NrmCVec /\ B e. X /\ A e. X ) -> ( B P A ) e. CC )
21 20 3adant3r1 
 |-  ( ( U e. NrmCVec /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( B P A ) e. CC )
22 1 3 dipcl 
 |-  ( ( U e. NrmCVec /\ C e. X /\ A e. X ) -> ( C P A ) e. CC )
23 22 3adant3r2 
 |-  ( ( U e. NrmCVec /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( C P A ) e. CC )
24 cjsub 
 |-  ( ( ( B P A ) e. CC /\ ( C P A ) e. CC ) -> ( * ` ( ( B P A ) - ( C P A ) ) ) = ( ( * ` ( B P A ) ) - ( * ` ( C P A ) ) ) )
25 21 23 24 syl2anc 
 |-  ( ( U e. NrmCVec /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( * ` ( ( B P A ) - ( C P A ) ) ) = ( ( * ` ( B P A ) ) - ( * ` ( C P A ) ) ) )
26 1 3 dipcj 
 |-  ( ( U e. NrmCVec /\ B e. X /\ A e. X ) -> ( * ` ( B P A ) ) = ( A P B ) )
27 26 3adant3r1 
 |-  ( ( U e. NrmCVec /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( * ` ( B P A ) ) = ( A P B ) )
28 1 3 dipcj 
 |-  ( ( U e. NrmCVec /\ C e. X /\ A e. X ) -> ( * ` ( C P A ) ) = ( A P C ) )
29 28 3adant3r2 
 |-  ( ( U e. NrmCVec /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( * ` ( C P A ) ) = ( A P C ) )
30 27 29 oveq12d 
 |-  ( ( U e. NrmCVec /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( ( * ` ( B P A ) ) - ( * ` ( C P A ) ) ) = ( ( A P B ) - ( A P C ) ) )
31 25 30 eqtrd 
 |-  ( ( U e. NrmCVec /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( * ` ( ( B P A ) - ( C P A ) ) ) = ( ( A P B ) - ( A P C ) ) )
32 11 31 sylan 
 |-  ( ( U e. CPreHilOLD /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( * ` ( ( B P A ) - ( C P A ) ) ) = ( ( A P B ) - ( A P C ) ) )
33 10 19 32 3eqtr3d 
 |-  ( ( U e. CPreHilOLD /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( A P ( B M C ) ) = ( ( A P B ) - ( A P C ) ) )
34 5 33 sylan2 
 |-  ( ( U e. CPreHilOLD /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A P ( B M C ) ) = ( ( A P B ) - ( A P C ) ) )